venomx8e0b

2023-03-13

Let f be a function so that (below). Which must be true?

I. f is continuous at x=2

II. f is differentiable at x=2

III. The derivative of f is continuous at x=2

(A) I (B) II (C) I and II (D) I and III (E) II and III

I. f is continuous at x=2

II. f is differentiable at x=2

III. The derivative of f is continuous at x=2

(A) I (B) II (C) I and II (D) I and III (E) II and III

neizhojen50q

Beginner2023-03-14Added 3 answers

(C) Explanation: Noting that a function $$ is differentiable at a point $$ if

$$

the given information effectively is that $$ is differentiable at $$ and that $$.Now, looking at the statements:

I: True

A function's differentiability at a point implies its continuity at that point.

II: True

The given information matches the definition of differentiability at $$.

III: False

The derivative of a function is not necessarily continuous, a classic example being $$, which is differentiable at $$, but whose derivative has a discontinuity at $$.

$$

the given information effectively is that $$ is differentiable at $$ and that $$.Now, looking at the statements:

I: True

A function's differentiability at a point implies its continuity at that point.

II: True

The given information matches the definition of differentiability at $$.

III: False

The derivative of a function is not necessarily continuous, a classic example being $$, which is differentiable at $$, but whose derivative has a discontinuity at $$.